We are concerned with a class of Kirchhoff type equations in $\mathbb{ R}^{ N}$ as follows: \begin{ equation*} \left\{ \begin{ array}{ ll} -M\left( \int_{ \mathbb{ R}^{ N}}|\nabla u|^{ 2}dx\right) \Delta u+\lambda V\left( x\right) u=f(x,u) & \text{ in }\mathbb{ R}^{ N}, \\ u\in H^{ 1}(\mathbb{ R}^{ N}), & \end{ array}% \right. \end{ equation*}% where $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{ R}^{ +},\mathbb{ R}^{ +})$, $V\in C(\mathbb{ R}^{ N},\mathbb{ R}^{ +})$ and $f\in C(\mathbb{ R}^{ N}\times \mathbb{ R}, \mathbb{ R})$ satisfying $\lim_{ |u|\rightarrow \infty }f(x,u) /|u|^{ k-1}=q(x)$ uniformly in $x\in \mathbb{ R}^{ N}$ for any $2<k<2^{ \ast}$($2^{ \ast}=\infty$ for $N=1,2$ and $2^{ \ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.
Categories: math.AP